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Two short proofs of the formula $\sum\limits_{n = 0}^\infty{{1 \over {{{(2n + 1)}^2}}} = {{{\pi ^2}} \over 8}} $

Published online by Cambridge University Press:  24 February 2022

Lubomir Markov
Lubomir Markov*
Affiliation:
Department of Mathematics and CS, Barry University, 11300 NE Second Avenue, Miami Shores, FL 33161, USA email: lmarkov@barry.edu
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Euler’s striking proof of the fact that the infinite sum

${1 \over {{1^2}}}\, + \,{1 \over {{2^2}}}\, + \,{1 \over {{3^2}}} + \,...\, + \,{1 \over {{n^2}}} + \,...\, = {{{\pi ^2}} \over 6}$
continues to charm all who become acquainted with it as well as to motivate the search for new proofs. With the introduction of the zeta function $\zeta (s)\zeta= \sum\limits_{n = 1}^\infty{{1 \over {{n^s}}}} $, the above sum is rewritten as $\zeta (2)$ and the problem of its evaluation in closed form is often named after Basel - the city of Euler’s birth. Many solutions to the Basel problem exist today, with several original ones appearing on the pages of The Mathematical Gazette. The purpose of this Note is to present two short proofs, the first of which is a simplification of one in the literature and the second is believed to be new, of the equivalent fact that
$\sum\limits_{n = 0}^\infty{{1 \over {{{(2n + 1)}^2}}} = {{{\pi ^2}} \over 8}} .$

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Type
Articles
Information
The Mathematical Gazette , Volume 106 , Issue 565 , March 2022 , pp. 28 - 31
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Copyright
© The Authors, 2022. Published by Cambridge University Press on behalf of The Mathematical Association

References

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